This is a typical symmetric polynomial factorization problem. $$x + y = 0 \implies f(x,y,z)=0 \implies (x+y)(y+z)(z+x) \mid f(x, y, z)$$  Letting $f(x,y,z)=(x+y)(y+z)(z+x)(m(x^2+y^2 + z^2)+n(xy+yz + zx))$ Setting $x=y=z=1$ and $x=y=1, z=-2$ leads to $m=n=5$ $\therefore\ f(x,y,z)=\boxed{5(x+y)(y+z)(z+x)(x^2+y^2+z^2+xy+yz+zx)}$